A new technique of second quantization for fermions

Edward R. Pike

Yuli V. Nazarov, Jeroen Danon

In this chapter we describe a technique to deal with identical particles that is called second quantization . Despite being a technique, second quantization helps a lot in understanding physics. One can learn and endlessly repeat newspaper-style statements particles are fields , fields are particles without grasping their meaning. Second quantization brings these concepts to an operational level. Since the procedure of second quantization is slightly different for fermions and bosons, we have to treat the two cases separately. We start with considering a system of identical bosons, and introduce creation and annihilation operators (CAPs) that change the number of particles in a given many-particle state, thereby connecting different states in Fock space. We show how to express occupation numbers in terms of CAPs, and then get to the heart of the second quantization: the commutation relations for the bosonic CAPs. As a next step, we construct field operators out of the CAPs and spend quite some time (and energy) presenting the physics of the particles in terms of these field operators. Only afterward, do we explain why all this activity has been called second quantization. In Section 3.4 we then present the same procedure for fermions, and we mainly focus on what is different as compared to bosons. The formalism of second quantization is summarized in Table 3.1.

Jonuzi, Leon

The Quantum-Mechanical description of subatomic particles is the most accurate theory of these particle’s behavior. The first quantization is the standard method for solving the Schrodinger equation, but the dimension of the partial differential equation increases rapidly, especially when dealing with more than one particle. These differential equations become difficult to solve and, although there exist methods in this formalism, a new method is approached. This thesis starts with an introduction where the first quantization is discussed in multi-particle physics and electronic systems. The second quantization formalism is then introduced and discussed with some definitions and notation. Some applications of the new formalism are then discussed, such as some spin operators and the number operator. Some new models and operators are introduced, such as the tight-binding model, the Hubbard model, and the number operator. Some numerical applications are also discussed for these models and operators. The second quantization formalism is then implemented in Python, and some analysis is performed for the tight-binding and Hubbard model. In other words, this thesis focuses on the implementation of the second quantization formalism for fermions in Python and is then used to analyze two models. It is also a first step into the multi-particle quantum mechanics.